3D turtle geometry: artwork, theory, program equivalence and symmetry

We define a 3D variant of turtle graphics and present the theoretical foundations of 3D turtle geometry. This theory enables one to reason about open and closed 3D polygonal paths by means of algebraic calculations. In particular, we introduce several equivalence relations on turtle programs and theorems that define corresponding standard forms. We also express the relationship between the symmetries of a 3D polygonal path and the symmetries of a generating turtle program in a suitable standard form. Finally, we discuss software tool support for 3D turtle geometry. Along the way, we present some artworks designed through 3D turtle graphics. These artworks have never been described in the literature before.